Charles & everyone,
Earlier peasthope wrote, "...changing a few words of a sentence can make a concept obvious rather than nebulous". Revise that to "obvious rather than difficult".
From: Charles Wells <charles@abstractmath.org> Date: Fri, 22 Apr 2011 09:37:44 -0500
Can you give specific examples? I suspect that in most cases the change introduces a useful metaphor that was hidden before.
Here is a small example from the _Conceptual Mathematics_ of Lawvere and Schanuel. No offense to the authors or the book. It's an indispensible and invaluable resource.
L&S page 292, "Definition ... equalizer ... and for each x:T-->X ... there is exactly one e:T-->E ... ." "For all T" is implicit.
http://en.wikipedia.org/wiki/Equalizer_(Mathematics) , "In category theory ... defined by a universal property, ... object E and morphism eq ... such
Hi, Peter, Actually, the word "other" below introduces a red herring: there is in fact every reason *not* to wish to restrict attention only to objects O *other* than E or T or X -- indeed, I can imagine that there might be settings in which there are *no* objects "other than" E or T or X, in which case the Wikipedia verbiage quoted paints you into a corner you really *don't* want to be in :-) . Cheers, -- Fred ------ Original Message ------ Received: Sat, 30 Apr 2011 03:30:49 PM EDT From: peasthope@shaw.ca To: categories@mta.ca Cc: peasthope@shaw.ca Subject: categories: Re: Explanations that,
given any other object O and morphism m ... ."
For me, the reference to "any other object O" helps. The definition in the Wikipedia seems to reveal the "universality" of the equalizer better. The diagram also helps.
A trivial issue for most readers but a small detail can make a difference for a student.
Regards, ... Peter E.
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